| import Mathlib | |
| import OAI.Probability.Ballisticity.Model | |
| namespace OAI | |
| section | |
| section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal Topology Classical | |
| namespace DirectionalTransience | |
| lemma integral_bind_nonneg_of_integrable {Ω G : Type*} [MeasurableSpace Ω] [MeasurableSpace G] | |
| (μ : Measure Ω) (Q : Ω → Measure G) (hQ : Measurable Q) | |
| (F : G → ℝ) (hF : Measurable F) (h0 : ∀ u, 0 ≤ F u) | |
| (hi : ∀ ω, Integrable F (Q ω)) : | |
| (∫ u, F u ∂μ.bind Q) = ∫ ω, ∫ u, F u ∂Q ω ∂μ := by | |
| have he (ω : Ω) : (∫ u, F u ∂Q ω) = (∫⁻ u, ENNReal.ofReal (F u) ∂Q ω).toReal := | |
| integral_eq_lintegral_of_nonneg_ae (Eventually.of_forall h0) hF.aestronglyMeasurable | |
| rw [integral_eq_lintegral_of_nonneg_ae (Eventually.of_forall h0) hF.aestronglyMeasurable, | |
| Measure.lintegral_bind hQ.aemeasurable hF.ennreal_ofReal.aemeasurable] | |
| simp_rw [he] | |
| symm | |
| apply integral_toReal | |
| · exact ((Measure.measurable_lintegral (hF.ennreal_ofReal)).comp hQ).aemeasurable | |
| · exact Eventually.of_forall fun ω => (hasFiniteIntegral_iff_ofReal (Eventually.of_forall h0)).mp (hi ω).hasFiniteIntegral | |
| end DirectionalTransience | |
| end | |
| end | |
| end OAI | |